Chapter 4.P1, Problem 2P

Denote by $G\left(i\omega \right)$ the frequency response of $\text{Eq}\text{.}\left(\text{i}\right)$, that is, the ratio of the steady-state response to the harmonic input $u\left(t\right)=A{e}^{i\omega t}$. The transmissibility $T_R$ of the platform mounting system is then defined to be the corresponding gain function, $T_R=\left|G\left(i\omega \right)\right|$. Show that $T_R$ can be expressed as

$T_R=\frac{\sqrt{1+{\left(2\zeta \frac{\omega }{{\omega }_0}\right)}^2}}{\sqrt{{\left[1-{\left(\frac{\omega }{{\omega }_0}\right)}^2\right]}^2+{\left[2\zeta \frac{\omega }{{\omega }_0}\right]}^2}}$.

Eq. $\left(\text{i}\right)$ $y^{″}+2\zeta {\omega }_0{y}^{\prime }+{\omega }_0^{2}y=2\zeta {\omega }_0{u}^{\prime }+{\omega }_0^{2}u$.